arXiv subjects

Ali Tamoussit

Publications and source records attributed to Ali Tamoussit.

4 recordsLinked to original sources

arXiv2026

Local properties of integral domains under extensions and pullback constructions

For a property $\mathcal{X}$ of integral domains, an integral domain $D$ is said to be a {\it locally $\mathcal{X}$-domain} if $D_P$ has the property $\mathcal{X}$ for every prime ideal $P$ of $D$. In this paper, we study the transfer of local properties of integral domains under several extensions and constructions, including flat overrings, Nagata ideal transforms, polynomial rings and their quotient extensions, and pullback constructions.

math.AC↗

arXiv2024

On the Krull dimension of rings of integer-valued rational functions

Let $D$ be an integral domain with quotient field $K$ and $E$ a subset of $K$. The \textit{ring of integer-valued rational functions on} $E$ is defined as $$\mathrm{int}_R(E,D):=\lbrace \varphi \in K(X);\; \varphi(E)\subseteq D\rbrace.$$ The main goal of this paper is to investigate the Krull dimension of the ring $\mathrm{int}_R(E,D).$ Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.

math.AC↗

arXiv2024

On the transfer of certain ring-theoretic properties in Anderson rings

Let $R$ be a commutative ring with unity and let $X$ be an indeterminate over $R$. The \textit{Anderson ring} of $R$ is defined as the quotient ring of the polynomial ring $R[X]$ by the set of polynomials that evaluate to $1$ at $0$. Specifically, the Anderson ring of $R$ is $R[X]_A$, where $A=\{f\in R[X]\mid f(0)=1\}$. In this paper, we aim to investigate the transfer of various ring-theoretic properties between the ring $R$ and its Anderson ring $R[X]_A$. Interesting results are established, accompanied by applications and illustrative examples.

math.AC↗

arXiv2024

On rings of integer-valued rational functions

Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the concepts of integer-valued polynomials, defined as $ Int^R_B(E,D) \:=\lbrace f \in B(X);\; f(E)\subseteq D\rbrace.$ The notion of $Int^R_B(E,D)$ boils down to the usual notion of integer-valued rational functions when the subset $E$ is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from $Int^R(E,D)$ to $D$.

math.AC↗